Abstract
In this paper, we show that special instances of parameterized NP-hard problems are as difficult as the general instances in terms of their subexponential time computability. For example, we show that the Planar Dominating Set problem on degree-3 graphs can be solved in \(2^{o(\sqrt{k})} p(n)\) parameterized time if and only if the general Planar Dominating Set problem can. Apart from their complexity theoretic implications, our results have some interesting algorithmic implications as well.
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References
Alber, J., Bodlaender, H.L., Ferneau, H., Niedermeier, R.: Fixed parameter algorithms for Dominating Set and related problems on planar graphs. Algorithmica 33, 461–493 (2002)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Buss, J., Goldsmith, J.: Nondeterminism within P. SIAM Journal on Computing 22, 560–572 (1993)
Cai, L., Juedes, D.: On the existence of subexponential parameterized algorithms. Journal of Computer and System Sciences 67(4), 789–807 (2003)
Chen, J., Chor, B., Fellows, M.R., Huang, X., Juedes, D.W., Kanj, I.A., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. Information and Computation 201(2), 216–231 (2005)
Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: lower bounds and upper bounds on kernel size. SIAM Journal on Computing 37(4), 1077–1106 (2007)
Chen, J., Huang, X., Kanj, I.A., Xia, G.: Linear FPT reductions and computational lower bounds. In: STOC 2004, pp. 212–221 (2004)
Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong Computational Lower Bounds via Parameterized Complexity. Journal of Computer and System Sceiences 72(8), 1346–1367 (2006)
Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. Journal of Algorithms 41, 280–301 (2001)
Chen, J., Kanj, I.A., Xia, G.: Improved Parameterized Upper Bounds for Vertex Cover. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 238–249. Springer, Heidelberg (2006)
Diestel, R.: Graph Theory. Springer, New York (1996)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fomin, F., Thilikos, D.: Dominating sets in planar graphs: branch-width and exponential speed-up. SIAM Journal on Computing 36(2), 281–309 (2006)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman, San Francisco (1979)
Johnson, D., Szegedy, M.: What are the least tractable instances of max independent set? In: SODA 1999, pp. 927–928 (1999)
Kanj, I.A., Perkovic, L.: Improved parameterized algorithms for planar dominating set. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 399–410. Springer, Heidelberg (2002)
Mahajan, M., Raman, V.: Parameterizing above guaranteed values: Max-Sat and Max-Cut. Journal of Algorithms 31, 335–354 (1999)
Nemhauser, G.L., Trotter, L.E.: Vertex packing: structural properties and algorithms. Mathematical Programming 8, 232–248 (1975)
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Chen, J., Kanj, I.A., Xia, G. (2009). On Parameterized Exponential Time Complexity. In: Chen, J., Cooper, S.B. (eds) Theory and Applications of Models of Computation. TAMC 2009. Lecture Notes in Computer Science, vol 5532. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02017-9_20
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DOI: https://doi.org/10.1007/978-3-642-02017-9_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02016-2
Online ISBN: 978-3-642-02017-9
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